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Topic: Fundamental period for complex functions
Replies: 3   Last Post: Aug 16, 2013 10:11 PM

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Peter Percival

Posts: 2,623
Registered: 10/25/10
Re: Fundamental period for complex functions
Posted: Aug 12, 2013 11:00 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply wrote:
> Suppose f from the complexes to the complexes is periodic. Then, is
> there something like a fundamental period for f?
> On the reals, if f is periodic then f may have a fundamental period,
> defined, when it exists, as the smallest period of f. By definition,
> periods are positive, and, if P is the set of periods of f, then the
> fundamental period is p*= infimum P, if p* > 0. In this case, we can
> show p* is in P. So, in this case, p* is the minimum of P. And it's
> known that if f is continuous, periodic and non constant, then f has
> a fundamental period (aka minimum period).
> In the complexes, the above definition doesn't make sense, but maybe
> we can define p* as the period with minimum absolute value, provided
> it's positive. So, the definition might be p = minimum {|p| : p is
> period of f}. Does this exist?
> For example, 2?i is a period of f(z) = e^z. Does f have a period with
> positive absolute value < 2??
> If we define p* for the complexes in this way, then, supposing f is
> continuous, periodic and non constant, then does such p* exist?
> Thank you

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